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Theorem nmoofval 30781
Description: The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoofval.1 𝑋 = (BaseSet‘𝑈)
nmoofval.2 𝑌 = (BaseSet‘𝑊)
nmoofval.3 𝐿 = (normCV𝑈)
nmoofval.4 𝑀 = (normCV𝑊)
nmoofval.6 𝑁 = (𝑈 normOpOLD 𝑊)
Assertion
Ref Expression
nmoofval ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
Distinct variable groups:   𝑥,𝑡,𝑧,𝑈   𝑡,𝑊,𝑥,𝑧   𝑡,𝑋,𝑧   𝑡,𝑌,𝑥   𝑡,𝐿   𝑡,𝑀
Allowed substitution hints:   𝐿(𝑥,𝑧)   𝑀(𝑥,𝑧)   𝑁(𝑥,𝑧,𝑡)   𝑋(𝑥)   𝑌(𝑧)

Proof of Theorem nmoofval
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoofval.6 . 2 𝑁 = (𝑈 normOpOLD 𝑊)
2 fveq2 6906 . . . . . 6 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 nmoofval.1 . . . . . 6 𝑋 = (BaseSet‘𝑈)
42, 3eqtr4di 2795 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54oveq2d 7447 . . . 4 (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑m 𝑋))
6 fveq2 6906 . . . . . . . . . . 11 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
7 nmoofval.3 . . . . . . . . . . 11 𝐿 = (normCV𝑈)
86, 7eqtr4di 2795 . . . . . . . . . 10 (𝑢 = 𝑈 → (normCV𝑢) = 𝐿)
98fveq1d 6908 . . . . . . . . 9 (𝑢 = 𝑈 → ((normCV𝑢)‘𝑧) = (𝐿𝑧))
109breq1d 5153 . . . . . . . 8 (𝑢 = 𝑈 → (((normCV𝑢)‘𝑧) ≤ 1 ↔ (𝐿𝑧) ≤ 1))
1110anbi1d 631 . . . . . . 7 (𝑢 = 𝑈 → ((((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))))
124, 11rexeqbidv 3347 . . . . . 6 (𝑢 = 𝑈 → (∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))))
1312abbidv 2808 . . . . 5 (𝑢 = 𝑈 → {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))})
1413supeq1d 9486 . . . 4 (𝑢 = 𝑈 → sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ))
155, 14mpteq12dv 5233 . . 3 (𝑢 = 𝑈 → (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )) = (𝑡 ∈ ((BaseSet‘𝑤) ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
16 fveq2 6906 . . . . . 6 (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
17 nmoofval.2 . . . . . 6 𝑌 = (BaseSet‘𝑊)
1816, 17eqtr4di 2795 . . . . 5 (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
1918oveq1d 7446 . . . 4 (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑m 𝑋) = (𝑌m 𝑋))
20 fveq2 6906 . . . . . . . . . . 11 (𝑤 = 𝑊 → (normCV𝑤) = (normCV𝑊))
21 nmoofval.4 . . . . . . . . . . 11 𝑀 = (normCV𝑊)
2220, 21eqtr4di 2795 . . . . . . . . . 10 (𝑤 = 𝑊 → (normCV𝑤) = 𝑀)
2322fveq1d 6908 . . . . . . . . 9 (𝑤 = 𝑊 → ((normCV𝑤)‘(𝑡𝑧)) = (𝑀‘(𝑡𝑧)))
2423eqeq2d 2748 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 = ((normCV𝑤)‘(𝑡𝑧)) ↔ 𝑥 = (𝑀‘(𝑡𝑧))))
2524anbi2d 630 . . . . . . 7 (𝑤 = 𝑊 → (((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))))
2625rexbidv 3179 . . . . . 6 (𝑤 = 𝑊 → (∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))))
2726abbidv 2808 . . . . 5 (𝑤 = 𝑊 → {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))})
2827supeq1d 9486 . . . 4 (𝑤 = 𝑊 → sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ))
2919, 28mpteq12dv 5233 . . 3 (𝑤 = 𝑊 → (𝑡 ∈ ((BaseSet‘𝑤) ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )) = (𝑡 ∈ (𝑌m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
30 df-nmoo 30764 . . 3 normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
31 ovex 7464 . . . 4 (𝑌m 𝑋) ∈ V
3231mptex 7243 . . 3 (𝑡 ∈ (𝑌m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )) ∈ V
3315, 29, 30, 32ovmpo 7593 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 normOpOLD 𝑊) = (𝑡 ∈ (𝑌m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
341, 33eqtrid 2789 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  m cmap 8866  supcsup 9480  1c1 11156  *cxr 11294   < clt 11295  cle 11296  NrmCVeccnv 30603  BaseSetcba 30605  normCVcnmcv 30609   normOpOLD cnmoo 30760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sup 9482  df-nmoo 30764
This theorem is referenced by:  nmooval  30782  hhnmoi  31920
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